chapter4
Elsa Marques
11/22/2017
Clustering and classification
This chapter will teach you how one can cluster and classify datasets. There is different methods to perform the clustering of the data, but in a very simple point of view you cluster what is similar. For that you can use different parameters, from simple concepts of means to more complex concepts like data populations distances.
About this week working data set “Boston”
The data set Boston is included in the library {MASS} and refers to the “Housing Values in Suburbs of Boston”. The data compiled in the dataset orginates from the following papers: Harrison and Rubinfeld (1978) & Belsley et al 1980
1. The data to be used in this exercise will be loaded from the library {MASS}.
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## nasaThe Boston data set includes the Housing Values in Suburbs of Boston (from 1978 - 1980). The information included in this data set includes values regarding Boston city crime, nitrogen oxide concentrations, average rooms per dwelling, lower status of the population, pupil-teacher ratio in town, etc..
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...## [1] 506 14You can see that the data includes a total of 506 observations distributed into the following 14 variables: - crim = per capita crime rate by town. - zn = proportion of residential land zoned for lots over 25,000 sq.ft. - indus = proportion of non-retail business acres per town. - chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise). - nox = nitrogen oxides concentration (parts per 10 million). - rm = average number of rooms per dwelling. - age = proportion of owner-occupied units built prior to 1940. - dis = weighted mean of distances to five Boston employment centres. - rad = index of accessibility to radial highways. - tax = full-value property-tax rate per $10,000. - ptratio = pupil-teacher ratio by town. - black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town. - lstat = lower status of the population (percent). - medv = median value of owner-occupied homes in $1000s.
2. Lets have a graphical overview of the data
- From the graphical overview using the ggpairs between the variables you can start by observing which of the variables are normally distributed and which of them presented skewed data. In this data set most of the variables presented themselves skewed in either direction, being the rm (average number of rooms per dwelling) the only variable that presents a normal distribution.
2.1 For a better understanding of the data lets also have a look on the summaries of the variables included in this dataset.
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00- From the variables summary you can mainly check the ranges of the data in the different variables. The skweded distrubition of most variables is now noticeable when you compare the Mean values and the minimum and maximum values of the data.
2.2 For a better understanding on how the variables relate to each other, lets visualize a correlation plot.
## corrplot 0.84 loaded## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00
By drawing the correlation plot you can see some variables are highly positive or negatively correlated with each other.
High positive numbers (Blue) mean that variables value will shift together in the same direction. For example, when you look at the crime data, you can see its higly positvely correlated with the tax, rad and lstat variables. Meaning that the higher the index of acessibility to radial highways, the higher value of property-tax rate per $10.000, and the amount of low status of the population the bigger amount of crime you can witness on those areas.
On the other side, you can also observe some High negative numbers (Red) meaning that while the value of one variable increases the other one will decrease. For example, you can see how lstat (percentage of the lower status of the population) inversely correlates with medv (median value of owner-occupied homes in $1000s) and rm (average number of rooms per dwelling). Meaning that the higher amount of low status population the less amount of rooms per dwelling, and the less amount of owner occupied homes with the value of $1000.
3. Standardizing the dataset and explore the scaled data summary and how variables change.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865## [1] "matrix"- When scaling the data, you just change the scale from which the data distributes in the x axis, not changing the values of the data itself. One can think this process is like grabbing the axis at the two ends and stretching or compressing it in order to make all your variables vary in the same comparable range, making it easier to compare. The scaling takes in account the mean (average) and standard deviation of the data.
3.1 Create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate). Quantiles from the old “crim”" variable will be used as the break points for the new “crime”" categorical variable.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610## crime
## low med_low med_high high
## 127 126 126 127- The new crime variable is now easier to interpret the amount of data fiiting in the the new assigned labels: “low”, “med_low”, “med_high”, “high”. You can also verify how the distribution between this categories is quite even.
3.2 Since the new crime category looks more informative, let’s drop the old crim variable from the dataset.
## # A tibble: 506 x 14
## # Groups: crime [4]
## zn indus chas nox rm age
## * <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.28454827 -1.2866362 -0.2723291 -0.1440749 0.4132629 -0.11989477
## 2 -0.48724019 -0.5927944 -0.2723291 -0.7395304 0.1940824 0.36680343
## 3 -0.48724019 -0.5927944 -0.2723291 -0.7395304 1.2814456 -0.26554897
## 4 -0.48724019 -1.3055857 -0.2723291 -0.8344581 1.0152978 -0.80908783
## 5 -0.48724019 -1.3055857 -0.2723291 -0.8344581 1.2273620 -0.51067434
## 6 -0.48724019 -1.3055857 -0.2723291 -0.8344581 0.2068916 -0.35080997
## 7 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3880270 -0.07015919
## 8 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.1603069 0.97784057
## 9 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.9302853 1.11638970
## 10 0.04872402 -0.4761823 -0.2723291 -0.2648919 -0.3994130 0.61548134
## # ... with 496 more rows, and 8 more variables: dis <dbl>, rad <dbl>,
## # tax <dbl>, ptratio <dbl>, black <dbl>, lstat <dbl>, medv <dbl>,
## # crime <fctr>- Now you can easily explore how the other variables vary according to the high or low crime rate category, and to have a better understanding on that I group the variables by crime categories.
3.3 Divide the dataset to train and test sets, so that 80% of the data belongs to the train set.
## [1] 506## zn indus chas nox rm
## 353 2.08538800 -1.37701059 -0.2723291 -1.24005818 -0.570202978
## 176 -0.48724019 -1.03300497 -0.2723291 -0.38570897 0.371988664
## 283 0.37030254 -1.13795583 3.6647712 -0.96476788 1.936140650
## 100 -0.48724019 -1.20209248 -0.2723291 -0.94664532 1.610216351
## 359 -0.48724019 1.01499462 3.6647712 1.85803641 -0.224353176
## 337 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.591551731
## 53 0.41317968 -0.80123846 -0.2723291 -0.99842406 0.322174907
## 431 -0.48724019 1.01499462 -0.2723291 0.25289548 0.090185122
## 430 -0.48724019 1.01499462 -0.2723291 1.07272553 0.135729129
## 180 -0.48724019 -1.26477147 -0.2723291 -0.57556435 0.989679257
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## 60 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 15 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
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## 225 -0.48724019 -0.71961001 -0.2723291 -0.43748771 2.819979033
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## 37 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 345 1.87100232 -1.07236154 -0.2723291 -0.61008351 0.838814735
## 439 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.497617217
## 477 -0.48724019 1.01499462 -0.2723291 0.51178918 0.283747151
## 141 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.157460416
## 124 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.610053984
## 201 3.58608778 -1.40907891 -0.2723291 -1.30909650 1.210283041
## 35 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 173 -0.48724019 -1.03300497 -0.2723291 -0.38570897 -1.014257045
## 233 -0.48724019 -0.71961001 -0.2723291 -0.41159834 2.921029798
## 465 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.107646658
## 496 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.874778524
## 10 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 460 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.289822685
## 121 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.590128481
## 94 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.104800158
## 56 3.37170210 -1.44552019 -0.2723291 -1.30909650 1.372533565
## 50 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 350 1.22784527 -1.44114723 -0.2723291 -1.08472196 0.931325998
## 449 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.141804663
## 307 0.92770532 -1.30558569 -0.2723291 -0.71364099 1.615909352
## 143 -0.48724019 1.23072696 3.6647712 2.72964520 -1.254786331
## 130 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.921745781
## 21 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 140 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.190195170
## 263 0.37030254 -1.04466617 -0.2723291 0.79657225 3.007848061
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 244 0.79907391 -0.90473168 -0.2723291 -1.09335175 0.154231381
## 425 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.024219796
## 252 0.45605682 -0.76917014 -0.2723291 -1.06746238 0.218277641
## 85 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.148538381
## 248 0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.083451404
## 246 0.45605682 -0.76917014 -0.2723291 -1.06746238 -0.967289788
## 59 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 322 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.130036128
## 227 -0.48724019 -0.71961001 -0.2723291 -0.43748771 2.498324485
## 340 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.426454706
## 326 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.208314890
## 377 -0.48724019 1.01499462 -0.2723291 1.00368721 0.518583436
## 346 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.385180450
## 301 2.51415937 -1.29683979 -0.2723291 -1.33498587 0.834544984
## 291 2.94293073 -0.90181638 -0.2723291 -1.24005818 0.820312482
## 440 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.935978283
## 208 -0.48724019 -0.07970124 -0.2723291 -0.56693456 -0.713951249
## 89 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.028107013
## 380 -0.48724019 1.01499462 -0.2723291 1.00368721 -0.087721155
## 268 0.37030254 -1.04466617 -0.2723291 0.17522737 2.864099790
## 127 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.955903786
## 194 2.08538800 -1.19626187 -0.2723291 -1.32635608 0.733494219
## 148 -0.48724019 1.23072696 -0.2723291 2.72964520 -1.933676683
## 104 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 349 2.94293073 -1.33036576 -0.2723291 -1.03294322 0.498657933
## 429 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.130418661
## 57 3.15731641 -1.51548743 -0.2723291 -1.24868797 0.139998879
## 409 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.950210785
## 321 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.201198639
## 467 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.473421963
## 398 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.765188257
## 468 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.400836202
## 134 -0.48724019 1.56744433 -0.2723291 0.59808708 -0.658444491
## 270 0.37030254 -0.60882854 3.6647712 -0.78267931 -0.518965970
## 406 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.856276271
## 305 0.92770532 -1.30558569 -0.2723291 -0.71364099 1.354031312
## 111 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 389 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.999146193
## age dis rad tax ptratio
## 353 -1.778931675 3.2840499862 -0.6373311 0.01639310 -0.07184181
## 176 -1.260260608 -0.3147359550 -0.5224844 -0.66594918 -0.85708096
## 283 -0.670538709 0.6728643674 -0.5224844 -1.14062207 -1.64232012
## 100 -0.215813389 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 359 0.526667797 -0.5092546567 1.6596029 1.52941294 0.80577843
## 337 -0.791325122 0.6819824315 -0.5224844 -1.09315478 0.80577843
## 53 -1.686565595 1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 431 0.622586419 -0.8274371034 1.6596029 1.52941294 0.80577843
## 430 0.960077867 -0.8675660836 1.6596029 1.52941294 0.80577843
## 180 -0.361467593 -0.4587728745 -0.7521778 -1.27709053 -0.30279450
## 323 -0.663433626 0.4397838527 -0.5224844 -0.71934988 0.52863520
## 60 -0.759352248 1.4897384367 -0.1779443 -0.73715011 0.57482574
## 15 0.565745754 0.3166899868 -0.6373311 -0.60068166 1.17530274
## 95 0.309962761 -0.0855021237 -0.6373311 -0.82021787 -0.11803234
## 314 0.505352547 -0.2527616128 -0.6373311 -0.61848189 -0.02565127
## 74 -2.215894287 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 285 -1.697223220 1.6679680968 -0.9818712 -0.73121670 -1.45755797
## 164 0.899684660 -0.7755306237 -0.5224844 -0.03107419 -1.73470120
## 382 1.084416821 -1.0807958128 1.6596029 1.52941294 0.80577843
## 72 -1.814457091 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512 0.34387304
## 296 -1.331311439 1.0283263992 -0.6373311 -0.70748306 -1.13422420
## 259 1.116389695 -0.9025186628 -0.5224844 -0.85581834 -2.51994036
## 2 0.366803426 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 146 1.116389695 -1.1283332201 -0.5224844 -0.03107419 -1.73470120
## 367 0.810871121 -0.9700968153 1.6596029 1.52941294 0.80577843
## 388 0.743372832 -1.0811757321 1.6596029 1.52941294 0.80577843
## 300 -2.080897708 1.9151531169 -0.5224844 -0.29807769 -1.68851066
## 280 -1.292233482 0.1451564051 -0.5224844 -1.14062207 -1.64232012
## 13 -1.050660656 0.7863652700 -0.5224844 -0.57694801 -1.50374851
## 269 -0.567515004 -0.4383522101 -0.5224844 -0.85581834 -2.51994036
## 131 1.077311738 -0.7961887377 -0.6373311 0.17066179 1.26768382
## 299 -1.722091011 1.9151531169 -0.5224844 -0.29807769 -1.68851066
## 342 -0.684748875 1.5400302593 -0.9818712 -0.73715011 -1.36517689
## 9 1.116389695 1.0861216287 -0.5224844 -0.57694801 -1.50374851
## 69 -1.128816570 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 370 1.002708366 -1.1579669285 1.6596029 1.52941294 0.80577843
## 19 -1.135921653 0.0006920764 -0.6373311 -0.60068166 1.17530274
## 487 0.398776300 -0.1183176566 1.6596029 1.52941294 0.80577843
## 278 -1.455650394 0.5070770657 -0.6373311 -0.91515245 -0.39517558
## 163 1.052443947 -0.8331358935 -0.5224844 -0.03107419 -1.73470120
## 390 1.077311738 -0.9815893753 1.6596029 1.52941294 0.80577843
## 413 1.116389695 -1.0643168114 1.6596029 1.52941294 0.80577843
## 414 1.116389695 -1.0474578907 1.6596029 1.52941294 0.80577843
## 195 -1.768274051 1.1514202651 -0.9818712 -0.84988492 -1.31898635
## 339 -1.111053862 0.4830471675 -0.5224844 -1.09315478 0.80577843
## 33 0.476932215 0.0925850666 -0.6373311 -0.60068166 1.17530274
## 273 -0.350809969 0.0581548764 -0.7521778 -1.09908819 0.06672981
## 475 0.952972784 -0.6483526248 1.6596029 1.52941294 0.80577843
## 470 -0.421860800 -0.4612898402 1.6596029 1.52941294 0.80577843
## 446 0.931657534 -0.8582105699 1.6596029 1.52941294 0.80577843
## 96 -0.382782843 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 87 -0.833955621 0.3002109855 -0.7521778 -0.95668632 0.02053927
## 67 -1.331311439 1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 417 0.789555872 -0.9381835908 1.6596029 1.52941294 0.80577843
## 99 -1.125264029 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 324 0.203386515 0.4397838527 -0.5224844 -0.71934988 0.52863520
## 284 -1.555121557 0.9925190015 -0.9818712 -1.24742347 -2.24279713
## 174 0.551535588 -0.5455369536 -0.5224844 -0.66594918 -0.85708096
## 113 0.864159245 -0.6846349217 -0.4076377 0.14099473 -0.30279450
## 381 0.828633829 -1.1295679579 1.6596029 1.52941294 0.80577843
## 488 -0.546199754 -0.3052379716 1.6596029 1.52941294 0.80577843
## 343 -0.315284553 1.1738829960 -0.9818712 0.08166062 -1.18041474
## 14 -0.240681180 0.4333252240 -0.6373311 -0.60068166 1.17530274
## 224 0.434301716 -0.2483450505 -0.1779443 -0.60068166 -0.48755665
## 40 -1.661697804 0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 372 1.116389695 -1.2470580136 1.6596029 1.52941294 0.80577843
## 316 0.324172928 0.0712146037 -0.6373311 -0.61848189 -0.02565127
## 422 0.949420242 -0.9120166463 1.6596029 1.52941294 0.80577843
## 466 -0.723826832 -0.3459843207 1.6596029 1.52941294 0.80577843
## 462 0.704294875 -0.5831489682 1.6596029 1.52941294 0.80577843
## 106 0.999155824 -0.8016975682 -0.5224844 -0.14380900 1.12911220
## 479 0.999155824 -0.7714939807 1.6596029 1.52941294 0.80577843
## 294 -1.782484217 0.8109650472 -0.6373311 -0.70748306 -1.13422420
## 115 0.555088129 -0.7306526517 -0.4076377 0.14099473 -0.30279450
## 125 0.967182950 -0.8494724251 -0.8670245 -1.30675758 0.29768250
## 416 1.116389695 -0.9309651233 1.6596029 1.52941294 0.80577843
## 364 0.725610124 -0.8977221812 1.6596029 1.52941294 0.80577843
## 261 0.469827132 -0.7992280924 -0.5224844 -0.85581834 -2.51994036
## 210 1.116389695 0.0379716616 -0.6373311 -0.77868399 0.06672981
## 331 -1.292233482 0.9871051509 -0.6373311 0.12912791 -0.71850935
## 385 0.803766038 -1.1186452769 1.6596029 1.52941294 0.80577843
## 411 1.116389695 -1.1312301050 1.6596029 1.52941294 0.80577843
## 84 -0.777114956 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 177 -0.759352248 -0.1140435641 -0.5224844 -0.66594918 -0.85708096
## 225 0.345488177 -0.4277144686 -0.1779443 -0.60068166 -0.48755665
## 186 0.007996729 -0.2447358168 -0.7521778 -1.27709053 -0.30279450
## 256 -1.757616426 2.5764502168 -0.9818712 -0.55321437 -0.94946204
## 105 0.761135540 -0.6525317376 -0.5224844 -0.14380900 1.12911220
## 38 -0.961847117 0.0660856927 -0.5224844 -0.76681717 0.34387304
## 275 -1.267365691 0.1341862342 -0.6373311 -0.91515245 -0.39517558
## 441 0.846396537 -0.9160057994 1.6596029 1.52941294 0.80577843
## 22 0.732715207 0.1031753182 -0.6373311 -0.60068166 1.17530274
## 142 1.116389695 -1.1186927669 -0.6373311 0.17066179 1.26768382
## 7 -0.070159185 0.8384142195 -0.5224844 -0.57694801 -1.50374851
## 62 0.881921953 1.4358373805 -0.1779443 -0.73715011 0.57482574
## 423 0.675874542 -0.8756393696 1.6596029 1.52941294 0.80577843
## 266 -0.205155765 -0.8588754287 -0.5224844 -0.85581834 -2.51994036
## 75 -2.222999370 0.2167712006 -0.5224844 -0.06074124 0.11292035
## 397 0.974288033 -1.0059517029 1.6596029 1.52941294 0.80577843
## 79 -0.528437047 0.5789293108 -0.5224844 -0.06074124 0.11292035
## 129 1.073759197 -0.9415078850 -0.6373311 0.17066179 1.26768382
## 403 1.116389695 -1.0239028917 1.6596029 1.52941294 0.80577843
## 348 -1.452097852 2.2511442825 -0.6373311 -0.33961157 -0.25660396
## 245 0.281542429 1.9755128019 -0.2927910 -0.46421320 0.29768250
## 444 1.116389695 -0.8627221120 1.6596029 1.52941294 0.80577843
## 442 1.016918532 -0.8215483536 1.6596029 1.52941294 0.80577843
## 237 0.281542429 0.1676191361 -0.1779443 -0.60068166 -0.48755665
## 325 -1.011582699 0.4397838527 -0.5224844 -0.71934988 0.52863520
## 162 0.789555872 -0.8662838558 -0.5224844 -0.03107419 -1.73470120
## 435 0.938762618 -0.7469416934 1.6596029 1.52941294 0.80577843
## 144 1.116389695 -1.1317999841 -0.5224844 -0.03107419 -1.73470120
## 172 1.020471073 -0.6678709809 -0.5224844 -0.03107419 -1.73470120
## 336 -1.210525026 1.0401513886 -0.5224844 -1.09315478 0.80577843
## 160 1.116389695 -0.9636381865 -0.5224844 -0.03107419 -1.73470120
## 293 -1.604857139 0.6276539660 -0.6373311 -0.96855315 0.34387304
## 183 0.839291454 -0.5197499285 -0.7521778 -1.27709053 -0.30279450
## 83 -1.292233482 0.7625253315 -0.6373311 -0.75495035 0.25149196
## 132 1.034681240 -0.7237666137 -0.6373311 0.17066179 1.26768382
## 480 0.690084708 -0.8756393696 1.6596029 1.52941294 0.80577843
## 464 0.757582998 -0.4717851119 1.6596029 1.52941294 0.80577843
## 64 -0.894348827 1.9878601804 -0.1779443 -0.73715011 0.57482574
## 234 0.064837394 -0.0679783442 -0.1779443 -0.60068166 -0.48755665
## 264 0.920999910 -0.8150422349 -0.5224844 -0.85581834 -2.51994036
## 471 0.547983046 -0.3617034834 1.6596029 1.52941294 0.80577843
## 354 -1.153684361 3.9566021965 -0.5224844 -1.31269099 -0.67231881
## 137 0.885474494 -0.8681834525 -0.6373311 0.17066179 1.26768382
## 393 1.009813449 -0.9616910999 1.6596029 1.52941294 0.80577843
## 271 -0.940531867 0.3010658040 -0.7521778 -1.09908819 0.06672981
## 481 -0.137657475 -0.1761128861 1.6596029 1.52941294 0.80577843
## 168 0.377461051 -0.6502047316 -0.5224844 -0.03107419 -1.73470120
## 5 -0.510674339 1.0766711351 -0.7521778 -1.10502160 0.11292035
## 191 -1.672355429 1.2749890302 -0.5224844 -0.06074124 -1.50374851
## 383 1.116389695 -1.0517319833 1.6596029 1.52941294 0.80577843
## 92 0.189176348 -0.3338269018 -0.8670245 -0.82021787 -0.30279450
## 70 -1.263813149 1.2836321952 -0.6373311 -0.37521203 0.20530143
## 289 -0.816192913 1.6726695986 -0.4076377 -0.68374941 -0.85708096
## 272 -1.857087590 0.3010658040 -0.7521778 -1.09908819 0.06672981
## 161 0.853501620 -0.9482039634 -0.5224844 -0.03107419 -1.73470120
## 356 -1.743406260 3.2248775491 -0.6373311 -0.44047956 1.63720813
## 363 0.981393116 -0.8032647354 1.6596029 1.52941294 0.80577843
## 221 0.707847416 -0.4432436716 -0.1779443 -0.60068166 -0.48755665
## 247 -1.196314860 2.0232876588 -0.2927910 -0.46421320 0.29768250
## 415 1.116389695 -1.0147848276 1.6596029 1.52941294 0.80577843
## 445 0.995603282 -0.9020437636 1.6596029 1.52941294 0.80577843
## 25 0.906789743 0.2871037683 -0.6373311 -0.60068166 1.17530274
## 76 -0.837508162 0.3360183832 -0.5224844 -0.06074124 0.11292035
## 482 0.224701764 -0.2200410597 1.6596029 1.52941294 0.80577843
## 30 0.665216917 0.2108349609 -0.6373311 -0.60068166 1.17530274
## 277 -0.695406500 0.4711746882 -0.6373311 -0.91515245 -0.39517558
## 361 0.690084708 -0.6063715378 1.6596029 1.52941294 0.80577843
## 48 0.601271169 0.8996287230 -0.7521778 -1.03975408 -0.25660396
## 338 -0.318837095 0.8642962245 -0.5224844 -1.09315478 0.80577843
## 128 0.974288033 -0.9530004450 -0.6373311 0.17066179 1.26768382
## 133 1.041786323 -0.6969823003 -0.6373311 0.17066179 1.26768382
## 253 -2.194579038 2.4275218358 -0.2927910 -0.46421320 0.29768250
## 73 -2.159053622 0.7086717651 -0.6373311 -0.61254848 0.34387304
## 156 0.498247464 -0.9733261297 -0.5224844 -0.03107419 -1.73470120
## 365 0.508905089 -0.8977221812 1.6596029 1.52941294 0.80577843
## 312 -0.560409921 -0.5483863487 -0.6373311 -0.61848189 -0.02565127
## 355 -1.658145262 3.2248775491 -0.6373311 -0.44047956 1.63720813
## 504 0.796660955 -0.7729186782 -0.9818712 -0.80241764 1.17530274
## 3 -0.265548971 0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 192 -1.341969064 1.2749890302 -0.5224844 -0.06074124 -1.50374851
## 402 1.116389695 -1.0547238481 1.6596029 1.52941294 0.80577843
## 109 1.013365990 -0.6468804374 -0.5224844 -0.14380900 1.12911220
## 55 -0.745142082 1.6738568465 -0.7521778 0.36053095 1.22149328
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## 305 -0.976057283 0.1077818401 -0.2927910 -1.10502160 -0.02565127
## 111 -0.503569256 -0.4830877123 -0.5224844 -0.14380900 1.12911220
## 389 1.116389695 -1.0474104008 1.6596029 1.52941294 0.80577843
## black lstat medv crime
## 353 0.390558315 -0.681000049 -0.427613550 low
## 176 0.375551994 -1.025486649 0.746669127 low
## 283 0.223407620 -1.350368321 2.551585092 low
## 100 0.440615895 -0.905056374 1.159842661 low
## 359 0.424514223 -0.164270149 0.018178948 high
## 337 0.440615895 -0.399529291 -0.329756660 low
## 53 0.440615895 -1.032488409 0.268257666 low
## 431 -2.992764527 0.698346703 -0.873406047 high
## 430 -3.241738006 1.600173411 -1.417055434 high
## 180 0.440615895 -1.066096858 1.594762170 low
## 323 0.440615895 -0.693603218 -0.231899770 med_high
## 60 0.440615895 -0.480749709 -0.318883672 med_low
## 15 0.255720500 -0.335113097 -0.471105500 med_high
## 95 0.440615895 -0.288901480 -0.210153795 low
## 314 0.402169045 -0.665596177 -0.101423917 med_high
## 74 0.224502972 -0.716008850 0.094289862 med_low
## 285 0.416737224 -0.672597937 1.051112783 low
## 164 0.348058662 -1.306957408 2.986504601 med_high
## 382 0.440615895 1.180067802 -1.264833606 high
## 72 0.221983663 -0.388326475 -0.090550930 med_low
## 494 0.440615895 -0.090051492 -0.079677942 med_low
## 296 0.440615895 -0.893853558 0.659685225 med_low
## 259 0.291538506 -0.681000049 1.464286318 med_high
## 2 0.440615895 -0.491952525 -0.101423917 low
## 146 -2.012862748 2.121104367 -0.949516961 med_high
## 367 -0.445195159 0.188618563 -0.068804954 high
## 388 0.440615895 2.707851869 -1.645388177 high
## 300 0.163272803 -1.108107419 0.703177176 low
## 280 0.440615895 -1.092703547 1.366429428 med_low
## 13 0.370513376 0.428078760 -0.090550930 med_low
## 269 0.368322672 -1.329363040 2.279760398 med_high
## 131 0.420242350 -0.007430722 -0.362375623 med_high
## 299 0.126688050 -1.075899322 -0.003567028 low
## 342 0.416956295 -1.003081017 1.105477722 low
## 9 0.328123258 2.419379350 -0.655946292 med_low
## 69 0.440615895 0.061186528 -0.558089402 med_low
## 370 0.204348498 -1.249542974 2.986504601 high
## 19 -0.741378303 -0.134862757 -0.253645746 med_high
## 487 0.394392046 0.325853062 -0.373248611 high
## 278 0.402826256 -1.189327837 1.148969673 low
## 163 0.360764744 -1.503006692 2.986504601 med_high
## 390 0.440615895 1.147859705 -1.199595679 high
## 413 -3.591483857 3.041135653 -0.503724464 high
## 414 -1.595971828 1.040032598 -0.677692268 high
## 195 0.219354818 -1.158520092 0.714050163 low
## 339 0.432291221 -0.580174703 -0.210153795 low
## 33 -1.359047220 2.108501199 -1.014754888 med_high
## 273 0.419366069 -0.689402161 0.203019739 med_low
## 475 -0.044844052 0.768364304 -0.949516961 high
## 470 0.440615895 0.295045317 -0.264518733 high
## 446 -3.435177145 1.586169891 -1.166976716 high
## 96 0.014304949 -0.840640181 0.637939249 med_low
## 87 0.430648193 0.028978431 -0.003567028 low
## 67 0.440615895 -0.337913801 -0.340629648 low
## 417 -3.670568261 1.839633609 -1.634515189 high
## 99 0.403702537 -1.271948607 2.312379361 low
## 324 0.377414092 -0.127860997 -0.438486537 med_high
## 284 0.425500039 -1.329363040 2.986504601 low
## 174 0.425280969 -0.505956045 0.116035838 med_low
## 113 0.419256534 0.498096362 -0.405867574 med_low
## 381 0.440615895 0.638131565 -1.319198544 high
## 488 0.345539353 -0.168471205 -0.210153795 high
## 343 0.364598476 -0.560569774 -0.655946292 low
## 14 0.440615895 -0.615183504 -0.231899770 med_high
## 224 0.440615895 -0.707606738 0.822780041 med_high
## 40 0.426704926 -1.166922204 0.898890955 low
## 372 0.103795196 -0.437338796 2.986504601 high
## 316 0.435358206 -0.161469445 -0.688565255 med_low
## 422 -0.401928760 0.426678408 -0.906025010 high
## 466 -0.243979021 0.206823139 -0.286264709 med_high
## 462 0.380700148 0.279641445 -0.525470439 high
## 106 0.409398367 0.534505515 -0.329756660 med_low
## 479 0.252215374 0.752960432 -0.862533059 high
## 294 0.440615895 -0.570372239 0.148654801 med_low
## 115 0.351235183 -0.308506409 -0.438486537 med_low
## 125 0.248710248 0.689944590 -0.405867574 med_low
## 416 -3.608352275 2.296148371 -1.667134152 high
## 364 -0.039805433 0.278241093 -0.623327329 high
## 261 0.395706469 -0.428936683 1.225080587 med_high
## 210 0.440615895 1.461538560 -0.275391721 med_high
## 331 0.130302712 -0.498954285 -0.297137697 low
## 385 -0.775991422 2.517403992 -1.493166348 high
## 411 -3.878356510 -0.356118378 -0.819041108 high
## 84 0.372046868 -0.720209906 0.039924923 low
## 177 0.400416482 -0.356118378 0.072543887 low
## 225 0.310816699 -1.192128541 2.421109239 med_high
## 186 0.333380947 0.069588640 0.768415102 low
## 256 0.421775843 -0.476548653 -0.177534831 low
## 105 0.394501581 -0.045240227 -0.264518733 med_low
## 38 0.440615895 -0.543765550 -0.166661844 low
## 275 0.440615895 -1.277550015 1.072858759 low
## 441 0.380919218 1.324304061 -1.308325557 high
## 22 0.392749018 0.164812578 -0.318883672 med_high
## 142 0.440615895 3.046737061 -0.884279035 med_high
## 7 0.426376321 -0.031236706 0.039924923 med_low
## 62 0.234470674 0.250234052 -0.710311231 med_low
## 423 -0.713337295 0.202622083 -0.188407819 high
## 266 0.391325061 -0.308506409 0.029051936 med_high
## 75 0.418927928 -0.822435605 0.170400776 low
## 397 0.440615895 0.940607604 -1.090865802 high
## 79 0.325603949 -0.043839875 -0.144915868 low
## 129 0.440615895 0.383267495 -0.492851476 med_high
## 403 0.212892242 1.072240695 -1.134357753 high
## 348 0.391653667 -0.881250390 0.061670899 low
## 245 0.173240505 -0.021434242 -0.536343427 med_low
## 444 0.329218610 0.867789298 -0.775549157 high
## 442 0.320784401 0.961612885 -0.590708366 high
## 237 0.348058662 -0.435938444 0.279130654 med_high
## 325 0.440615895 -0.914858839 0.268257666 med_high
## 162 0.194490331 -1.529613381 2.986504601 med_high
## 435 -2.804583076 0.352459751 -1.177849704 high
## 144 0.440615895 1.927855787 -0.753803182 high
## 172 -0.093587210 -0.087250788 -0.373248611 med_high
## 336 0.440615895 -0.650192305 -0.155788856 low
## 160 0.083640722 -0.737014131 0.083416874 med_high
## 293 0.440615895 -1.113708827 0.583574310 low
## 183 0.410165113 -1.096904603 1.670873085 med_low
## 83 0.440615895 -0.830837717 0.246511690 low
## 132 0.440615895 -0.055042691 -0.318883672 med_high
## 480 0.291867112 0.063987232 -0.123169893 high
## 464 0.406879058 -0.330912041 -0.253645746 high
## 64 0.426157250 -0.441539852 0.268257666 med_low
## 234 0.244000235 -1.218735230 2.801663810 med_high
## 264 0.402497650 -0.196478246 0.920636930 med_high
## 471 0.440615895 0.509299178 -0.286264709 high
## 354 0.304354123 -1.141715868 0.822780041 low
## 137 0.236332772 0.594720652 -0.558089402 med_high
## 393 0.440615895 1.824229736 -1.395309459 high
## 271 0.350249366 0.048583360 -0.155788856 med_high
## 481 0.440615895 -0.267896200 0.050797911 high
## 168 -1.413705278 -0.071846915 0.137781813 med_high
## 5 0.440615895 -1.025486649 1.486032293 low
## 191 0.230089266 -1.057694746 1.573016195 med_low
## 383 0.440615895 1.532956514 -1.221341655 high
## 92 0.403921608 -0.623585616 -0.057931966 low
## 70 0.440615895 -0.540964846 -0.177534831 med_low
## 289 0.440615895 -0.707606738 -0.025313003 low
## 272 0.440615895 -0.849042293 0.290003641 med_low
## 161 -0.194469117 -1.001680665 0.485717421 med_high
## 356 0.212125496 -0.991878200 -0.210153795 med_low
## 363 0.264154709 -0.344915562 -0.188407819 high
## 221 0.383657598 -0.412132459 0.453098458 med_high
## 247 0.367008250 -0.489151821 0.192146752 med_high
## 415 -2.939968567 3.406627533 -1.688880128 high
## 445 -1.272295352 1.559563202 -1.275706593 high
## 25 0.412465352 0.510699530 -0.753803182 med_high
## 76 0.290881295 -0.519959566 -0.123169893 med_low
## 482 0.398663919 -0.688001809 0.126908825 high
## 30 0.258020739 -0.094252548 -0.166661844 med_high
## 277 0.356821477 -0.924661303 1.159842661 med_low
## 361 0.195914288 -0.681000049 0.268257666 high
## 48 0.395049257 0.860787538 -0.645073304 med_low
## 338 0.417723041 -0.293102536 -0.438486537 low
## 128 0.388148541 0.635330861 -0.688565255 med_high
## 133 0.318593697 -0.214682823 0.050797911 med_high
## 253 0.322208358 -1.277550015 0.768415102 med_low
## 73 0.375004318 -0.998879961 0.029051936 med_low
## 156 -2.942816482 0.331454470 -0.753803182 med_high
## 365 -0.023265620 -1.031088057 -0.068804954 med_high
## 312 0.440615895 -0.934463767 -0.047058979 med_high
## 355 0.286171282 -0.644590896 -0.471105500 low
## 504 0.440615895 -0.982075736 0.148654801 low
## 3 0.396035074 -1.207532413 1.322937477 low
## 192 0.361860096 -1.115109179 0.866271992 low
## 402 0.440615895 1.073641047 -1.667134152 high
## 109 0.422433054 -0.053642339 -0.297137697 med_low
## 55 0.440615895 0.300646725 -0.394994586 low
## 119 -0.197645637 0.380466791 -0.231899770 med_low
## 489 0.420790026 0.757161488 -0.797295133 med_low
## 139 0.387381794 1.213676250 -1.003881900 med_low
## 185 0.375990135 0.185817859 0.420479494 med_low
## 499 0.440615895 0.037380543 -0.144915868 med_low
## 58 0.396801820 -1.218735230 0.985874857 low
## 463 0.440615895 0.187218211 -0.329756660 high
## 407 0.148376017 1.496547361 -1.156103728 high
## 352 0.154509988 -1.003081017 0.170400776 low
## 302 0.428019349 -0.441539852 -0.057931966 low
## 41 0.426595391 -1.494604580 1.344683452 low
## 357 0.230636942 0.692745294 -0.514597451 high
## 114 0.440615895 0.621327341 -0.416740562 med_low
## 401 0.440615895 1.976868108 -1.841101956 high
## 420 -3.376137680 1.412526239 -1.536658299 high
## 484 0.397020891 -0.312707465 -0.079677942 med_high
## 153 -0.146711775 -0.074647619 -0.786422145 med_high
## 386 0.440615895 2.542610329 -1.667134152 high
## 287 -0.165113687 0.038780895 -0.264518733 low
## 493 0.440615895 0.097595681 -0.264518733 med_low
## 369 0.206429666 -1.315359520 2.986504601 high
## 29 0.342472368 0.020576319 -0.449359525 med_high
## 42 0.314759966 -1.094103899 0.442225470 med_low
## 108 0.339733988 0.201221731 -0.231899770 med_low
## 11 0.392639483 1.091845624 -0.819041108 med_low
## 24 0.414765591 1.012025558 -0.873406047 med_high
## 206 0.440615895 -0.249691623 0.007305960 med_low
## 360 0.373142220 0.002371742 0.007305960 high
## 18 0.329437681 0.282442149 -0.547216415 med_high
## 81 0.440615895 -1.031088057 0.594447298 low
## 432 -3.015985986 0.985418869 -0.916897998 high
## 315 0.427362137 -0.472347596 0.137781813 med_high
## 297 0.396254144 -0.737014131 0.496590409 low
## 71 0.296358054 -0.830837717 0.181273764 med_low
## 170 -0.291736362 -0.186675782 -0.025313003 med_high
## 150 -0.052840120 1.231880827 -0.775549157 med_high
## 404 0.440615895 0.996621685 -1.547531287 high
## 303 0.295043632 -0.557769070 0.420479494 med_low
## 238 0.365803363 -1.109507771 0.975001869 med_high
## 318 0.440615895 0.460286857 -0.297137697 med_low
## 47 0.440615895 0.209623843 -0.275391721 med_low
## 187 0.393844370 -1.148717628 2.986504601 low
## 17 0.330533033 -0.850442645 0.061670899 med_high
## 304 0.369746629 -1.091303195 1.148969673 med_low
## 20 0.375442459 -0.192277190 -0.471105500 med_high
## 154 -1.037561447 0.439281577 -0.340629648 med_high
## 68 0.433057967 -0.637589136 -0.057931966 low
## 461 -1.111169093 0.527503755 -0.666819280 high
## 424 -3.879232792 1.489545601 -0.993008912 high
## 451 -3.903330533 0.670339662 -0.993008912 high
## 257 0.324946738 -1.336364800 2.334125337 low
## 501 0.440615895 0.234830180 -0.623327329 med_low
## 503 0.440615895 -0.500354637 -0.210153795 low
## 396 0.386724583 0.625528397 -1.025627875 high
## 211 0.400635552 0.646533677 -0.090550930 med_low
## 126 0.310488093 0.302047077 -0.123169893 med_low
## 368 -2.467324237 0.094794977 0.061670899 high
## 384 0.440615895 1.667390309 -1.112611777 high
## 1 0.440615895 -1.074498970 0.159527789 low
## 335 0.358464505 -0.826636661 -0.199280807 low
## 196 0.411370000 -1.355969729 2.986504601 low
## 44 0.413889309 -0.730012370 0.235638703 med_low
## 286 0.416737224 -0.619384560 -0.057931966 low
## 391 0.413560704 0.624128045 -0.808168120 high
## 203 0.423966547 -1.336364800 2.149284545 low
## 491 -0.418906714 2.384370549 -1.569277262 med_low
## 421 -0.415401588 0.331454470 -0.634200317 high
## 31 0.038293155 1.392921311 -1.069119826 med_high
## 167 0.138298780 -1.253744030 2.986504601 med_high
## 274 0.373470826 -0.850442645 1.377302416 med_low
## 497 0.440615895 1.188469914 -0.308010684 med_high
## 179 0.378947585 -0.802830676 0.801034065 low
## 250 0.406002776 -0.853243349 0.398733519 med_low
## 181 0.425938180 -0.713208146 1.877459852 low
## 243 0.176088420 -0.200679302 -0.036185991 med_low
## 469 0.132164810 0.766963952 -0.373248611 high
## 419 -3.726650277 1.115651608 -1.493166348 high
## 217 0.395706469 0.120001313 0.083416874 low
## 251 0.433824713 -0.945666583 0.203019739 med_low
## 436 -2.703591634 1.486744897 -0.993008912 high
## 136 0.416189548 0.603122764 -0.481978488 med_high
## 447 -0.423507192 0.719351983 -0.829914096 high
## 376 0.440615895 0.110198849 -0.819041108 high
## 169 -0.652654802 -0.217483527 0.137781813 med_high
## 320 0.433277038 0.010773855 -0.166661844 med_high
## 428 -3.700690438 0.261436868 -1.264833606 high
## 117 0.401183228 -0.085850436 -0.144915868 med_low
## 495 0.440615895 0.131204129 0.213892727 med_high
## 483 0.422871195 -0.790227508 0.268257666 high
## 212 0.422433054 1.586169891 -0.351502635 med_high
## 313 0.434043784 -0.130661701 -0.340629648 med_high
## 205 0.371061052 -1.368572897 2.986504601 low
## 86 0.390229709 -0.857444405 0.442225470 low
## 151 0.176636095 0.202622083 -0.112296905 med_high
## 282 0.389462963 -1.129112700 1.399048391 low
## 26 -0.583319029 0.540106923 -0.938643973 med_high
## 209 0.269960074 0.281041797 0.203019739 med_low
## 103 -3.131326538 -0.283300072 -0.427613550 med_low
## 433 -2.833938506 -0.087250788 -0.699438243 high
## 135 -1.028689097 0.652135085 -0.753803182 med_high
## 506 0.440615895 -0.668396881 -1.156103728 low
## 373 -0.096325589 -0.528361678 2.986504601 high
## 310 0.433386573 -0.375723306 -0.242772758 med_high
## 241 0.378728515 -0.178273670 -0.057931966 med_low
## 122 0.229979731 0.226428068 -0.242772758 low
## 182 0.440615895 -0.448541612 1.486032293 low
## 412 -3.522914830 1.199672730 -0.579835378 high
## 149 0.003460966 2.193922673 -0.514597451 med_high
## 486 0.349920761 -0.290301832 -0.144915868 med_high
## 219 0.440615895 0.737556559 -0.112296905 med_low
## 262 0.347182381 -0.755218707 2.236268447 med_high
## 123 0.234580209 0.738956911 -0.221026782 med_low
## 474 0.197228711 -0.139063813 0.790161078 high
## 98 0.440615895 -1.182326077 1.757856987 med_low
## 120 0.381466894 0.134004834 -0.351502635 med_low
## 82 0.426704926 -0.760820115 0.148654801 low
## 198 -0.025894464 -0.566171183 0.844526016 low
## 308 0.440615895 -0.717409202 0.616193274 low
## 218 0.395487398 -0.414933163 0.670558212 low
## 197 0.440615895 -1.200530653 1.170715648 low
## 32 0.219683424 0.054184768 -0.873406047 med_high
## 505 0.402826256 -0.864446166 -0.057931966 med_low
## 175 0.400416482 -0.421934923 0.007305960 med_low
## 455 -3.833666154 0.848184370 -0.829914096 high
## 226 0.277408467 -1.123511291 2.986504601 med_high
## 319 0.422104448 -0.321109577 0.061670899 med_high
## 12 0.440615895 0.086392864 -0.394994586 med_low
## 23 0.440615895 0.849584722 -0.797295133 med_high
## 213 0.375332924 0.472890025 -0.014440015 med_low
## 4 0.415751408 -1.360170785 1.181588636 low
## 61 0.421009097 0.069588640 -0.416740562 med_low
## 223 0.369308489 -0.381324714 0.540082359 med_high
## 45 0.358354970 -0.434538092 -0.144915868 med_low
## 434 -2.809402625 0.499496714 -0.895152022 high
## 152 -0.165113687 0.087793216 -0.318883672 med_high
## 258 0.361750561 -1.054894042 2.986504601 med_high
## 358 0.379714331 0.086392864 -0.090550930 high
## 222 0.422433054 1.233281179 -0.090550930 med_high
## 472 0.423418871 0.030378783 -0.318883672 high
## 260 0.386176907 -0.805631380 0.822780041 med_high
## 448 0.348825409 0.530304459 -1.079992814 high
## 490 -0.138277566 1.584769539 -1.688880128 med_low
## 341 0.440615895 -0.470947244 -0.416740562 low
## 458 -3.868498343 0.600322060 -0.982135924 high
## 220 0.406002776 -0.301504649 0.050797911 med_low
## 426 -3.822712635 1.643584324 -1.547531287 high
## 399 0.440615895 2.511802584 -1.906339882 high
## 459 -0.925178346 0.500897066 -0.829914096 high
## 306 0.401840439 -0.521359918 0.637939249 low
## 157 -2.936025300 0.488293898 -1.025627875 med_high
## 366 -0.021622592 -0.774823635 0.540082359 high
## 216 0.404797889 -0.445740908 0.268257666 med_low
## 102 0.426157250 -0.697804274 0.431352482 med_low
## 54 0.440615895 -0.591377519 0.094289862 low
## 190 0.440615895 -1.017084537 1.344683452 med_low
## 410 -1.942212553 0.998022037 0.540082359 high
## 193 0.370403840 -1.369973249 1.507778268 med_low
## 240 0.296905730 -0.739814835 0.083416874 med_low
## 473 0.401949974 0.239031236 0.072543887 med_high
## 452 -0.015160016 0.710949871 -0.797295133 high
## 27 0.221326452 0.302047077 -0.645073304 med_high
## 80 0.431414939 -0.497553933 -0.242772758 med_low
## 485 0.153962312 0.096195329 -0.210153795 med_high
## 166 -1.276238619 -0.398128939 0.268257666 med_high
## 8 0.440615895 0.909799859 0.496590409 med_low
## 178 0.426376321 -0.891052854 0.224765715 low
## 39 0.402607185 -0.353317674 0.235638703 med_low
## 232 0.213220848 -1.036689465 0.996747845 med_high
## 408 -0.269281649 -0.073247267 0.583574310 high
## 165 0.421009097 -0.141864517 0.018178948 med_high
## 438 -3.804748865 1.932056843 -1.504039336 high
## 281 0.335571651 -1.245341918 2.486347165 low
## 37 0.228774844 -0.174072614 -0.275391721 med_low
## 345 0.342800973 -1.126311996 0.942382906 low
## 439 -3.151590547 2.992123331 -1.536658299 high
## 477 0.433057967 0.843983314 -0.634200317 high
## 141 0.344005860 1.611376228 -0.927770986 med_high
## 124 0.149361834 1.786420232 -0.568962390 med_low
## 201 0.302601560 -1.148717628 1.127223698 low
## 35 -1.186967442 1.076441751 -0.982135924 med_high
## 173 0.440615895 0.285242853 0.061670899 med_low
## 233 0.320236725 -1.425987330 2.084046619 med_high
## 465 0.440615895 0.079391104 -0.123169893 high
## 496 0.401073693 0.692745294 0.061670899 med_low
## 10 0.328999540 0.622727693 -0.394994586 med_low
## 460 0.440615895 0.286643205 -0.275391721 high
## 121 0.355726125 0.240431588 -0.057931966 low
## 94 0.434372389 -0.902255670 0.268257666 low
## 56 0.429990982 -1.098304955 1.399048391 low
## 50 0.440615895 0.496696010 -0.340629648 med_low
## 350 0.363393588 -0.947066935 0.442225470 low
## 449 0.440615895 0.766963952 -0.916897998 high
## 307 0.440615895 -0.865846518 1.181588636 low
## 143 0.440615895 1.983869868 -0.993008912 med_high
## 130 0.440615895 0.796371345 -0.895152022 med_high
## 21 0.217930861 1.171665689 -0.971262937 med_high
## 140 0.440615895 0.813175569 -0.514597451 med_high
## 263 0.330642568 -0.944266231 2.856028748 med_high
## 107 0.427143067 0.841182610 -0.329756660 med_low
## 244 0.197557316 -1.045091578 0.126908825 med_low
## 425 -3.866855315 0.631129805 -1.177849704 high
## 252 0.223407620 -1.269147903 0.246511690 med_low
## 85 0.440615895 -0.424735627 0.148654801 low
## 248 0.213220848 -0.350516970 -0.221026782 med_low
## 246 0.355507055 0.813175569 -0.438486537 med_low
## 59 0.372485009 -0.811232788 0.083416874 med_low
## 322 0.440615895 -0.809832436 0.061670899 med_low
## 227 0.336338397 -1.333564096 1.638254121 med_high
## 340 0.440615895 -0.407931403 -0.384121599 low
## 326 0.405345565 -1.060495450 0.224765715 med_low
## 377 0.069510683 1.482543841 -0.938643973 high
## 346 0.317279275 -0.297303592 -0.547216415 low
## 301 0.374456642 -0.921860599 0.246511690 low
## 291 0.440615895 -1.305557056 0.648812237 low
## 440 0.440615895 1.432131167 -1.058246839 high
## 208 0.358793111 0.757161488 -0.003567028 med_low
## 89 0.440615895 -1.001680665 0.116035838 low
## 380 0.406002776 1.278092444 -1.340944520 high
## 268 0.305230404 -0.730012370 2.986504601 med_high
## 127 0.028654058 2.045485358 -0.742930194 med_high
## 194 0.401949974 -1.067497210 0.931509918 low
## 148 0.383767133 2.363365269 -0.862533059 med_high
## 104 0.413998845 0.110198849 -0.351502635 med_low
## 349 0.375332924 -0.933063415 0.213892727 low
## 429 -2.847301799 1.241683291 -1.253960618 high
## 57 0.440615895 -0.963871160 0.235638703 low
## 409 -0.460420549 1.925055083 -0.579835378 high
## 321 0.440615895 -0.763620819 0.137781813 med_low
## 467 -3.665748713 0.629729453 -0.384121599 high
## 398 0.398992524 1.017626966 -1.525785311 high
## 468 -0.278044464 1.213676250 -0.373248611 high
## 134 0.350687507 0.332854822 -0.449359525 med_high
## 270 0.379714331 0.139606242 -0.199280807 med_low
## 406 0.309940417 1.446134688 -1.906339882 high
## 305 0.405345565 -0.801430324 1.475159305 low
## 111 0.403264396 0.048583360 -0.090550930 med_low
## 389 0.177950518 2.516003640 -1.340944520 high## zn indus chas nox rm age
## 6 -0.4872402 -1.30558569 -0.2723291 -0.83445805 0.206891639 -0.35080997
## 16 -0.4872402 -0.43682573 -0.2723291 -0.14407485 -0.641365488 -0.42896588
## 28 -0.4872402 -0.43682573 -0.2723291 -0.14407485 -0.338213193 0.71850504
## 34 -0.4872402 -0.43682573 -0.2723291 -0.14407485 -0.830657767 0.93876262
## 36 -0.4872402 -0.75459363 -0.2723291 -0.48063666 -0.500463717 -0.01331852
## 43 -0.4872402 -0.61611679 -0.2723291 -0.92075595 -0.164576667 -2.20168412
## 46 -0.4872402 -0.61611679 -0.2723291 -0.92075595 -0.857699521 -1.23539282
## 49 -0.4872402 -0.61611679 -0.2723291 -0.92075595 -1.260479332 0.94942024
## 51 0.4131797 -0.80123846 -0.2723291 -0.99842406 -0.457766211 -0.81264037
## 52 0.4131797 -0.80123846 -0.2723291 -0.99842406 -0.241432178 -0.19805068
## 63 0.5846882 -0.87557866 -0.2723291 -0.87760700 0.243896145 -0.02752869
## 65 0.2631097 -1.42219777 -0.2723291 -1.19604625 1.166162284 -0.32238964
## 66 2.9429307 -1.13212523 -0.2723291 -1.35224545 0.007636609 -1.80379947
## 77 -0.4872402 0.24681257 -0.2723291 -1.01568364 -0.008019143 0.21049160
## 78 -0.4872402 0.24681257 -0.2723291 -1.01568364 -0.205850923 -0.80908783
## 88 -0.4872402 -0.96886832 -0.2723291 -0.91212616 -0.232892677 -0.41830826
## 90 -0.4872402 -1.12629463 -0.2723291 -0.56693456 1.130581029 -0.19449814
## 91 -0.4872402 -1.12629463 -0.2723291 -0.56693456 0.188389387 -0.08792189
## 93 0.7133196 0.56895343 -0.2723291 -0.78267931 0.223970642 -0.53198959
## 97 -0.4872402 -1.20209248 -0.2723291 -0.94664532 -0.173116168 0.03641706
## 101 -0.4872402 -0.37560439 -0.2723291 -0.29941107 0.629596953 0.40232884
## 110 -0.4872402 -0.37560439 -0.2723291 -0.29941107 -0.079181654 0.80376604
## 112 -0.4872402 -0.16424500 -0.2723291 -0.06640675 0.612517950 0.46272205
## 116 -0.4872402 -0.16424500 -0.2723291 -0.06640675 -0.507579968 0.69718979
## 118 -0.4872402 -0.16424500 -0.2723291 -0.06640675 -0.375217698 0.49824746
## 138 -0.4872402 1.56744433 -0.2723291 0.59808708 0.241049645 1.05954903
## 145 -0.4872402 1.23072696 -0.2723291 2.72964520 -1.966411438 1.03823378
## 147 -0.4872402 1.23072696 -0.2723291 2.72964520 -0.934555033 1.11638970
## 155 -0.4872402 1.23072696 3.6647712 2.72964520 -0.221506675 0.97428803
## 158 -0.4872402 1.23072696 -0.2723291 0.43412107 0.937018999 1.02402361
## 159 -0.4872402 1.23072696 -0.2723291 0.43412107 -0.311171439 1.11638970
## 171 -0.4872402 1.23072696 -0.2723291 0.43412107 -0.583012230 0.92455245
## 184 -0.4872402 -1.26477147 -0.2723291 -0.57556435 0.396183918 0.96007787
## 188 1.4422310 -1.12192167 -0.2723291 -1.01568364 0.707875715 -0.97605728
## 189 1.4422310 -1.12192167 -0.2723291 -1.01568364 0.386221166 -1.40236227
## 199 2.9429307 -1.40179066 -0.2723291 -1.30046671 1.408114820 -1.07552845
## 200 3.5860878 -1.40907891 -0.2723291 -1.30909650 0.982563006 -1.89261301
## 202 3.0501236 -1.32745046 -0.2723291 -1.20553902 -0.174539418 -1.07197591
## 204 3.5860878 -1.23270315 -0.2723291 -1.19604625 2.232176694 -1.25670807
## 207 -0.4872402 -0.07970124 -0.2723291 -0.56693456 0.058873617 -0.57106755
## 214 -0.4872402 -0.07970124 -0.2723291 -0.56693456 0.128612878 -1.28868094
## 215 -0.4872402 -0.07970124 -0.2723291 -0.56693456 -1.241977079 -2.08800279
## 228 -0.4872402 -0.71961001 -0.2723291 -0.43748771 1.250134047 0.40232884
## 229 -0.4872402 -0.71961001 -0.2723291 -0.43748771 1.994493909 -1.83221980
## 230 -0.4872402 -0.71961001 -0.2723291 -0.43748771 0.380528166 -1.67590797
## 231 -0.4872402 -0.71961001 -0.2723291 -0.43748771 -0.432147707 -0.01687106
## 235 -0.4872402 -0.71961001 3.6647712 -0.41159834 0.628173703 -0.07371173
## 236 -0.4872402 -0.71961001 -0.2723291 -0.41159834 -0.282706434 -0.25133880
## 239 0.7990739 -0.90473168 -0.2723291 -1.09335175 0.279477400 -1.77893168
## 242 0.7990739 -0.90473168 -0.2723291 -1.09335175 -0.269897182 -0.12344731
## 249 0.4560568 -0.76917014 -0.2723291 -1.06746238 0.211161390 -0.69185396
## 254 0.4560568 -0.76917014 -0.2723291 -1.06746238 2.810016281 -2.13773837
## 255 2.9429307 -1.09276866 -0.2723291 -1.40402419 -0.251394930 -1.29933856
## 265 0.3703025 -1.04466617 -0.2723291 0.79657225 1.311333806 0.81797620
## 267 0.3703025 -1.04466617 -0.2723291 0.79657225 1.038069765 0.56929830
## 276 1.2278453 -0.68899934 -0.2723291 -0.92938574 0.810349730 -0.91566408
## 279 1.2278453 -0.68899934 -0.2723291 -0.92938574 0.280900651 -1.29578602
## 288 1.7638095 -0.84788329 -0.2723291 -1.29183692 -0.107646658 -1.32420636
## 290 1.7638095 -0.84788329 -0.2723291 -1.29183692 0.399030418 -1.62261985
## 292 2.9429307 -0.90181638 -0.2723291 -1.24005818 1.228785293 -1.45209785
## 295 -0.4872402 0.40569652 -0.2723291 -1.01568364 -0.392296701 -0.93342678
## 298 -0.4872402 0.40569652 -0.2723291 -1.01568364 -0.703988498 -0.37567776
## 309 -0.4872402 -0.18027916 -0.2723291 -0.09229612 0.498657933 0.49469492
## 311 -0.4872402 -0.18027916 -0.2723291 -0.09229612 -1.866783923 -1.09329115
## 317 -0.4872402 -0.18027916 -0.2723291 -0.09229612 -0.527505471 0.51956271
## 327 -0.4872402 -0.54760720 -0.2723291 -0.53241540 0.038948114 -1.40946735
## 328 -0.4872402 -0.54760720 -0.2723291 -0.53241540 -0.286976185 -0.88369120
## 329 -0.4872402 -1.15107469 -0.2723291 -0.81719847 -0.592974981 -1.51959614
## 330 -0.4872402 -1.15107469 -0.2723291 -0.81719847 0.068836369 -1.82511472
## 332 1.0134596 -0.74001712 -0.2723291 -1.00791683 -0.823541516 -1.42723006
## 333 1.0134596 -0.74001712 -0.2723291 -1.00791683 -0.360985196 -1.60840968
## 334 -0.4872402 -0.86683276 -0.2723291 -0.34256002 0.044641115 -1.08263353
## 344 1.8710023 -1.07236154 -0.2723291 -0.61008351 0.585476196 -0.43251842
## 347 -0.4872402 -0.98344483 -0.2723291 -0.97253469 -0.550277475 -0.57817263
## 351 1.2278453 -1.44114723 -0.2723291 -1.08472196 0.292286652 -0.85882341
## 362 -0.4872402 1.01499462 -0.2723291 1.85803641 -0.047870149 0.80021350
## 371 -0.4872402 1.01499462 3.6647712 0.65849561 1.040916265 1.02757616
## 374 -0.4872402 1.01499462 -0.2723291 0.97779784 -1.962141687 1.11638970
## 375 -0.4872402 1.01499462 -0.2723291 0.97779784 -3.055197852 1.11638970
## 378 -0.4872402 1.01499462 -0.2723291 1.00368721 0.724954717 1.07375920
## 379 -0.4872402 1.01499462 -0.2723291 1.00368721 0.135729129 0.98139312
## 387 -0.4872402 1.01499462 -0.2723291 1.25395112 -2.323647242 1.11638970
## 392 -0.4872402 1.01499462 -0.2723291 1.25395112 -0.332520192 0.49469492
## 394 -0.4872402 1.01499462 -0.2723291 1.19354259 -0.130418661 0.85350162
## 395 -0.4872402 1.01499462 -0.2723291 1.19354259 -0.565933227 0.92810499
## 400 -0.4872402 1.01499462 -0.2723291 1.19354259 -0.615746985 0.32772547
## 405 -0.4872402 1.01499462 -0.2723291 1.19354259 -1.072610303 0.59771863
## 418 -0.4872402 1.01499462 -0.2723291 1.07272553 -1.395688102 0.72916267
## 427 -0.4872402 1.01499462 -0.2723291 0.25289548 -0.637095738 -0.31528455
## 437 -0.4872402 1.01499462 -0.2723291 1.59914271 0.251012396 0.87836941
## 443 -0.4872402 1.01499462 -0.2723291 1.59914271 -0.093414156 1.11638970
## 450 -0.4872402 1.01499462 -0.2723291 1.36613839 0.188389387 1.05599649
## 453 -0.4872402 1.01499462 -0.2723291 1.36613839 0.017599361 0.82508129
## 454 -0.4872402 1.01499462 -0.2723291 1.36613839 1.577481596 1.09152190
## 456 -0.4872402 1.01499462 -0.2723291 1.36613839 0.342100410 0.63679658
## 457 -0.4872402 1.01499462 -0.2723291 1.36613839 -0.439263958 0.68653217
## 476 -0.4872402 1.01499462 -0.2723291 0.25289548 -0.174539418 1.02402361
## 478 -0.4872402 1.01499462 -0.2723291 0.51178918 -1.395688102 1.02047107
## 492 -0.4872402 2.42017014 -0.2723291 0.46864023 -0.429301206 1.07375920
## 498 -0.4872402 -0.21088983 -0.2723291 0.26152527 -0.698295497 0.07194248
## 500 -0.4872402 -0.21088983 -0.2723291 0.26152527 -1.018526795 0.17496618
## 502 -0.4872402 0.11562398 -0.2723291 0.15796779 0.438881424 0.01865435
## dis rad tax ptratio black
## 6 1.076671135 -0.7521778 -1.10502160 0.11292035 0.41016511
## 16 0.334118787 -0.6373311 -0.60068166 1.17530274 0.42659539
## 28 0.312653344 -0.6373311 -0.60068166 1.17530274 -0.55089661
## 34 -0.003724486 -0.6373311 -0.60068166 1.17530274 0.02295823
## 36 -0.206458943 -0.5224844 -0.76681717 0.34387304 0.44061589
## 43 0.914588047 -0.7521778 -1.03975408 -0.25660396 0.29241479
## 46 0.619913109 -0.7521778 -1.03975408 -0.25660396 0.44061589
## 49 0.985395514 -0.7521778 -1.03975408 -0.25660396 0.44061589
## 51 1.434032764 -0.6373311 -0.98041997 -0.76469989 0.42593818
## 52 1.434032764 -0.6373311 -0.98041997 -0.76469989 0.40852209
## 63 1.629121344 -0.1779443 -0.73715011 0.57482574 0.44061589
## 65 2.577684955 -0.7521778 -1.14062207 0.06672981 0.40052602
## 66 1.337533251 -0.6373311 -0.42267932 -1.08803366 0.44061589
## 77 0.122123795 -0.5224844 -0.06074124 0.11292035 0.18605612
## 78 0.140312434 -0.5224844 -0.06074124 0.11292035 0.33173792
## 88 -0.022530493 -0.7521778 -0.95668632 0.02053927 0.42144724
## 90 -0.180719408 -0.8670245 -0.82021787 -0.30279450 0.43141494
## 91 -0.333731922 -0.8670245 -0.82021787 -0.30279450 0.38891529
## 93 -0.061329756 -0.6373311 -0.82021787 -0.11803234 0.41991374
## 97 -0.142395045 -0.8670245 -0.78461740 -0.21041342 0.38508156
## 101 -0.483087712 -0.5224844 -0.14380900 1.12911220 0.41717537
## 110 -0.593596750 -0.5224844 -0.14380900 1.12911220 0.37850944
## 112 -0.530720099 -0.4076377 0.14099473 -0.30279450 0.42626679
## 116 -0.632538482 -0.4076377 0.14099473 -0.30279450 -0.12885754
## 118 -0.497524647 -0.4076377 0.14099473 -0.30279450 0.41443699
## 138 -0.923794146 -0.6373311 0.17066179 1.26768382 0.40972697
## 145 -1.163095840 -0.5224844 -0.03107419 -1.73470120 0.44061589
## 147 -1.082030551 -0.5224844 -0.03107419 -1.73470120 -2.05273356
## 155 -0.971474023 -0.5224844 -0.03107419 -1.73470120 -0.39053710
## 158 -0.910734419 -0.5224844 -0.03107419 -1.73470120 0.07400163
## 159 -0.967722319 -0.5224844 -0.03107419 -1.73470120 -0.03049494
## 171 -0.650204732 -0.5224844 -0.03107419 -1.73470120 -0.70523169
## 184 -0.450224689 -0.7521778 -1.27709053 -0.30279450 0.44061589
## 188 -0.003059627 -0.5224844 -0.06074124 -1.50374851 0.40742673
## 189 0.366459420 -0.5224844 -0.06074124 -1.50374851 0.28660942
## 199 1.668775425 -0.8670245 -0.47014661 -2.70470251 0.38913436
## 200 1.832330701 -0.7521778 -0.03700760 -0.67231881 0.44061589
## 202 1.175355183 -0.8670245 -0.35741180 -1.73470120 0.40633138
## 204 0.628271335 -0.6373311 -1.09315478 -1.73470120 0.39548740
## 207 0.265875775 -0.6373311 -0.77868399 0.06672981 0.41838025
## 214 0.071404563 -0.6373311 -0.77868399 0.06672981 0.31914137
## 215 -0.098561851 -0.6373311 -0.77868399 0.06672981 -0.08482439
## 228 -0.275129364 -0.1779443 -0.60068166 -0.48755665 0.16874956
## 229 -0.199430436 -0.1779443 -0.60068166 -0.48755665 0.22822717
## 230 -0.199430436 -0.1779443 -0.60068166 -0.48755665 0.25922563
## 231 -0.058670320 -0.1779443 -0.60068166 -0.48755665 0.23742812
## 235 -0.067978344 -0.1779443 -0.60068166 -0.48755665 0.03862176
## 236 -0.067978344 -0.1779443 -0.60068166 -0.48755665 0.21990249
## 239 1.137315760 -0.4076377 -0.64221553 -0.85708096 0.24903885
## 242 1.206746019 -0.4076377 -0.64221553 -0.85708096 0.41564187
## 249 1.914535748 -0.2927910 -0.46421320 0.29768250 0.19755732
## 254 2.427521836 -0.2927910 -0.46421320 0.29768250 0.44061589
## 255 2.576450217 -0.9818712 -0.55321437 -0.94946204 0.39669229
## 265 -0.885659742 -0.5224844 -0.85581834 -2.51994036 0.34192469
## 267 -0.789350190 -0.5224844 -0.85581834 -2.51994036 0.30008225
## 276 0.224274608 -0.6373311 -0.91515245 -0.39517558 0.44061589
## 279 0.163962412 -0.6373311 -0.91515245 -0.39517558 0.44061589
## 288 1.672669599 -0.4076377 -0.68374941 -0.85708096 0.44061589
## 290 1.672669599 -0.4076377 -0.68374941 -0.85708096 0.16480630
## 292 0.627653966 -0.6373311 -0.96855315 0.34387304 0.44061589
## 295 0.810965047 -0.6373311 -0.70748306 -1.13422420 0.44061589
## 298 1.199100142 -0.6373311 -0.70748306 -1.13422420 0.44061589
## 309 -0.226784628 -0.6373311 -0.61848189 -0.02565127 0.44061589
## 311 -0.605801659 -0.6373311 -0.61848189 -0.02565127 -0.06817505
## 317 0.096669199 -0.6373311 -0.61848189 -0.02565127 0.37270408
## 327 0.769743799 -0.5224844 -0.71934988 0.52863520 0.44061589
## 328 0.769743799 -0.5224844 -0.71934988 0.52863520 0.44061589
## 329 0.674146595 -0.6373311 0.12912791 -0.71850935 0.28222802
## 330 0.674146595 -0.6373311 0.12912791 -0.71850935 0.20303408
## 332 1.351400307 -0.9818712 -0.61848189 -0.71850935 0.40906976
## 333 1.351400307 -0.9818712 -0.61848189 -0.71850935 0.06107647
## 334 1.264826188 -0.5224844 -1.09315478 0.80577843 0.36186010
## 344 0.919906918 -0.5224844 -0.22687676 -0.39517558 0.44061589
## 347 2.003389383 -0.7521778 -0.33367816 0.15911089 0.08692678
## 351 2.373098390 -0.9818712 -0.43454615 0.57482574 0.44061589
## 362 -0.712131584 1.6596029 1.52941294 0.80577843 -0.06598434
## 371 -1.231243871 1.6596029 1.52941294 0.80577843 0.38749133
## 374 -1.244636028 1.6596029 1.52941294 0.80577843 0.44061589
## 375 -1.262302277 1.6596029 1.52941294 0.80577843 0.44061589
## 378 -1.157349560 1.6596029 1.52941294 0.80577843 0.44061589
## 379 -1.144004893 1.6596029 1.52941294 0.80577843 0.44061589
## 387 -1.105490570 1.6596029 1.52941294 0.80577843 0.44061589
## 392 -0.772776208 1.6596029 1.52941294 0.80577843 0.23775673
## 394 -0.951623237 1.6596029 1.52941294 0.80577843 0.44061589
## 395 -0.955944820 1.6596029 1.52941294 0.80577843 0.44061589
## 400 -1.089723917 1.6596029 1.52941294 0.80577843 -0.20279379
## 405 -1.038909706 1.6596029 1.52941294 0.80577843 -0.29808940
## 418 -1.019866249 1.6596029 1.52941294 0.80577843 -2.51179552
## 427 -0.853604048 1.6596029 1.52941294 0.80577843 -3.63683142
## 437 -0.851229552 1.6596029 1.52941294 0.80577843 -3.60572343
## 443 -0.850184774 1.6596029 1.52941294 0.80577843 0.42736214
## 450 -0.764607943 1.6596029 1.52941294 0.80577843 -0.57466575
## 453 -0.677606414 1.6596029 1.52941294 0.80577843 0.31125484
## 454 -0.637477434 1.6596029 1.52941294 0.80577843 0.21026340
## 456 -0.645503230 1.6596029 1.52941294 0.80577843 -3.34908249
## 457 -0.576737829 1.6596029 1.52941294 0.80577843 -3.79204278
## 476 -0.754635060 1.6596029 1.52941294 0.80577843 -0.59054835
## 478 -0.804641943 1.6596029 1.52941294 0.80577843 -0.07879996
## 492 -0.915103491 -0.6373311 1.79641644 0.75958789 0.36624150
## 498 -0.428521797 -0.4076377 -0.10227512 0.34387304 0.44061589
## 500 -0.662552110 -0.4076377 -0.10227512 0.34387304 0.42823842
## 502 -0.625177545 -0.9818712 -0.80241764 1.17530274 0.38683412
## lstat medv crime
## 6 -1.042290874 0.67055821 low
## 16 -0.585776111 -0.28626471 med_high
## 28 0.647934029 -0.84078708 med_high
## 34 0.797771697 -1.02562788 med_high
## 36 -0.416333515 -0.39499459 low
## 43 -0.958269752 0.30087663 med_low
## 46 -0.342114857 -0.35150264 med_low
## 49 2.542610329 -0.88427903 med_low
## 51 0.111599201 -0.30801068 med_low
## 52 -0.451342316 -0.22102678 low
## 63 -0.829437365 -0.03618599 med_low
## 65 -0.644590896 1.13809669 low
## 66 -1.117909883 0.10516285 low
## 77 -0.095652900 -0.27539172 med_low
## 78 -0.333712745 -0.18840782 med_low
## 88 -0.589977167 -0.03618599 low
## 90 -0.973673624 0.67055821 low
## 91 -0.538164142 0.00730596 low
## 93 -0.629187024 0.03992492 low
## 97 -0.183875078 -0.12316989 med_low
## 101 -0.452742668 0.54008236 med_low
## 110 0.405673128 -0.34062965 med_high
## 112 -0.349116618 0.02905194 med_low
## 116 0.435080520 -0.46023251 med_low
## 118 -0.329511689 -0.36237562 med_low
## 138 0.271239333 -0.59070837 med_high
## 145 2.329756820 -1.16697672 med_high
## 147 0.559711851 -0.75380318 med_high
## 155 0.345457990 -0.60158135 med_high
## 158 -1.129112700 2.04055467 med_high
## 159 -0.871447926 0.19214675 med_high
## 171 0.248833700 -0.55808940 med_high
## 184 -0.976474328 1.08373175 med_low
## 188 -0.836439125 1.02936681 low
## 189 -1.133313756 0.79016108 med_low
## 199 -0.844841237 1.31206449 low
## 200 -1.133313756 1.34468345 low
## 202 -0.731412722 0.17040078 low
## 204 -1.238340158 2.82340979 low
## 207 -0.235688103 0.20301974 med_low
## 214 -0.458344076 0.60532029 med_low
## 215 2.366165973 0.12690883 med_high
## 228 -0.881250390 0.98587486 med_high
## 229 -1.222936286 2.62769601 med_high
## 230 -1.245341918 0.97500187 med_high
## 231 -0.140464165 0.19214675 med_high
## 235 -0.644590896 0.70317718 med_high
## 236 -0.248291271 0.15952779 med_high
## 239 -0.881250390 0.12690883 med_low
## 242 -0.035437762 -0.26451873 med_low
## 249 -0.438739148 0.21389273 med_low
## 254 -1.276149663 2.20364948 med_high
## 255 -0.851842997 -0.06880495 low
## 265 -0.637589136 1.51865126 med_high
## 267 0.299246373 0.88801797 med_high
## 276 -1.354569377 1.02936681 med_low
## 279 -0.765021171 0.71405016 low
## 288 -0.772022931 0.07254389 low
## 290 -0.440139500 0.24651169 low
## 292 -1.273348959 1.60563516 low
## 295 -0.315508169 -0.09055093 low
## 298 0.446283337 -0.24277276 med_low
## 309 -1.136114460 0.02905194 med_high
## 311 -0.001829314 -0.69943824 med_high
## 317 0.794970993 -0.51459745 med_high
## 327 -0.910657783 0.05079791 med_high
## 328 0.019175967 -0.03618599 med_low
## 329 -0.375723306 -0.35150264 low
## 330 -0.744015891 0.00730596 low
## 332 -0.031236706 -0.59070837 low
## 333 -0.675398641 -0.34062965 low
## 334 -0.976474328 -0.03618599 low
## 344 -0.766421523 0.14865480 low
## 347 0.002371742 -0.57983538 low
## 351 -0.934463767 0.03992492 low
## 362 0.215225251 -0.28626471 high
## 371 -1.357370081 2.98650460 high
## 374 3.097149734 -0.94951696 high
## 375 3.545262384 -0.94951696 high
## 378 1.202473434 -1.00388190 high
## 379 1.545559682 -1.02562788 high
## 387 2.188321265 -1.30832556 high
## 392 0.855186130 0.07254389 high
## 394 0.352459751 -0.94951696 high
## 395 0.517701290 -1.06911983 high
## 400 2.424980758 -1.76499104 high
## 405 2.062289582 -1.52578531 high
## 418 1.958663532 -1.31919854 high
## 427 0.425278056 -1.34094452 high
## 437 0.755761136 -1.40618245 high
## 443 0.551309739 -0.44935953 high
## 450 0.932205492 -1.03650086 high
## 453 0.646533677 -0.69943824 high
## 454 0.572315020 -0.51459745 high
## 456 0.766963952 -0.91689800 high
## 457 0.890194931 -1.06911983 high
## 476 1.602974115 -1.00388190 high
## 478 1.716402630 -1.14523074 high
## 492 0.758561840 -0.97126294 med_low
## 498 0.202622083 -0.46023251 med_high
## 500 0.342657286 -0.54721641 med_low
## 502 -0.417733867 -0.01444002 low4. Fit the linear discriminant analysis on the train set. Use the categorical crime rate (“crime”) as the target variable and all the other variables in the dataset as predictor variables. Draw the LDA (bi)plot.
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2425743 0.2524752 0.2475248 0.2574257
##
## Group means:
## zn indus chas nox rm
## low 0.94980017 -0.9105920 -0.07145661 -0.8586919 0.5003426
## med_low -0.09440995 -0.2727685 0.03646311 -0.5432534 -0.1364047
## med_high -0.38347752 0.2464336 0.23949396 0.4255776 0.1250548
## high -0.48724019 1.0170690 -0.04518867 1.0400319 -0.4061323
## age dis rad tax ptratio
## low -0.8673785 0.8473704 -0.6818635 -0.7705710 -0.42345550
## med_low -0.3439487 0.3491847 -0.5472552 -0.4838749 -0.04014242
## med_high 0.5038605 -0.4020077 -0.3984500 -0.2761834 -0.24413252
## high 0.8078993 -0.8389260 1.6386213 1.5144083 0.78135074
## black lstat medv
## low 0.38063197 -0.78908436 0.585460441
## med_low 0.31523354 -0.11739954 -0.001861461
## med_high 0.09192049 0.03266136 0.166703960
## high -0.82414899 0.82202202 -0.666610184
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.12398859 0.709619648 -1.01674042
## indus 0.02212879 -0.244881780 0.20550238
## chas -0.06708134 -0.068704666 0.09084434
## nox 0.37776875 -0.717917520 -1.22801750
## rm -0.08385034 -0.103393572 -0.19194512
## age 0.31422520 -0.457667401 -0.32036627
## dis -0.07689234 -0.418110954 0.20665628
## rad 2.97555794 1.066402249 -0.11116982
## tax 0.01712221 -0.214966607 0.60561793
## ptratio 0.10867461 0.005481813 -0.31536610
## black -0.12511645 -0.005148864 0.11994751
## lstat 0.15954535 -0.166417054 0.51332882
## medv 0.15496084 -0.333619601 -0.14143962
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9423 0.0420 0.0157
- According to the linear description fit you can see how most of the high crime category correlates with the ptratio (pupil-teacher ratio by town).
5.Save the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then predict the classes with the LDA model on the test data. Cross tabulate the results with the crime categories from the test set.
## [1] low med_high med_high med_high low med_low med_low
## [8] med_low med_low low med_low low low med_low
## [15] med_low low low low low med_low med_low
## [22] med_high med_low med_low med_low med_high med_high med_high
## [29] med_high med_high med_high med_high med_low low med_low
## [36] low low low low med_low med_low med_high
## [43] med_high med_high med_high med_high med_high med_high med_low
## [50] med_low med_low med_high low med_high med_high med_low
## [57] low low low low low med_low med_high
## [64] med_high med_high med_high med_low low low low
## [71] low low low low low high high
## [78] high high high high high high high
## [85] high high high high high high high
## [92] high high high high high high high
## [99] med_low med_high med_low low
## Levels: low med_low med_high high## predicted
## correct low med_low med_high high
## low 17 12 0 0
## med_low 3 16 5 0
## med_high 0 13 13 0
## high 0 0 0 23- You can see how the predicted and corrected data tend to agree on the most extreme values (“low” and “high”), in the “med-low” and “med-high” you can see the most discrepance between the correct data and the predicted one.
6.Reload the Boston dataset and standardize the dataset (we did not do this in the Datacamp exercises, but you should scale the variables to get comparable distances). Calculate the distances between the observations.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618- You can see how the two methods to calculate the distances between the variables , gives you as result different numbers, but you can also note how the trend between the variations is maintained.
6.1 Run k-means algorithm on the dataset. Investigate what is the optimal number of clusters and run the algorithm again. Visualize the clusters (for example with the pairs() or ggpairs() functions, where the clusters are separated with colors):
While checking for the optimal number of clusters you should look in the graph when defining the k max = 10 for the k value where the curve varies the most dramatically. That appears to be around k = 2
When exploring the pairs using the colors to separate the km clusters, you can see that the variable that better separetes the two clusters (Red Vs Black cluster) would be tax (full-value property-tax rate per $10,000) and rad (index of accessibility to radial highways). An almost complete cluster separation can also be observed when you check the crim (per capita crime rate by town) and black (1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town) variables.
7. Optional / Bonus sections:
7.1 Perform k-means on the original Boston data with some reasonable number of clusters (> 2). Remember to standardize the dataset.
For the reasonable amount of clusters (> 2) I select k = 3 as it was the following position in the curve you could observe a dramatically change.
Now when you check at the pairs with the clusters divided by colors, since we have more clusters it gets a bit more confusing to read the data. Nonetheless, after careful observation you the variables were you can more easily identify the different clusters (Red, Black and Green) are medv (median value of owner-occupied homes in $1000s) and partialy successful will be age (proportion of owner-occupied units built prior to 1940).
7.2 Then perform LDA using the clusters as target classes. Include all the variables in the Boston data in the LDA model. Visualize the results with a biplot (include arrows representing the relationships of the original variables to the LDA solution).
I was trying the following approach but I guess I missed something, further work/learning required in this section ; if the peer review knows how to fix this and explain me what is wrong I highly appreciate it!!
k-means clustering km <-kmeans(boston_scaled, centers = 3)
linear discriminant analysis lda.fit <- lda(km ~ . , data = boston_scaled$km)
print the lda.fit object lda.fit
function for the lda biplot arrows lda.arrows <- function(x, myscale = 2, arrow_heads = 0.1, color = “red”, tex = 0.75, choices = c(1,2)){ heads <- coef(x) arrows(x0 = 0, y0 = 0, x1 = myscale * heads[,choices[1]], y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads) text(myscale * heads[,choices], labels = row.names(heads), cex = tex, col=color, pos=3) }
target classes as numeric classes <- as.numeric(boston_scaled$km)
plot the lda results plot(lda.fit, dimen = 2, col = classes, pch = classes) lda.arrows(lda.fit, myscale = 5)
I got stuck in this section as unfortunetely I couldn’t manage to fix the errors in my code
8. Super-Bonus: Run the provided code for the (scaled) train data that you used to fit the LDA. The code creates a matrix product, which is a projection of the data points.
## [1] 404 13## [1] 13 3##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layout## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'hoverinfo', 'hoverlabel', 'stream', 'x', 'y', 'z', 'text', 'hovertext', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'error_x', 'error_y', 'error_z', 'scene', 'xcalendar', 'ycalendar', 'zcalendar', 'idssrc', 'customdatasrc', 'hoverinfosrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'textpositionsrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule'8.1 Draw another 3D plot where the color is defined by the clusters of the k-means. How do the plots differ?
- If I performed this correcly it seems that they don’t differ much. Maybe the clusters don’t separate that well the the data in the extreme negative x values (-2 < x < -4 ).